UNIFORM POLYNOMIAL STABILITY OF SECOND ORDER INTEGRO-DIFFERENTIAL EQUATIONS IN HILBERT SPACES WITH POSITIVE DEFINITE KERNELS

摘要

We are concerned with the polynomial stability and the integrabilityof the energy for second order integro-differential equations in Hilbertspaces with positive definite kernels, where the memory can be oscillating orsign-varying or not locally absolutely continuous (without any control conditionson the derivative of the kernel). For this stability problem, tools fromthe theory of existing positive definite kernels can not be applied. In orderto solve the problem, we introduce and study a new mathematical concept -generalized positive definite kernel (GPDK). With the help of GPDK and itsproperties, we obtain an effcient criterion of the polynomial stability for evolutionequations with such a general but more complicated and useful memory.Moreover, in contrast to existing positive definite kernels, GPDK allows us todirectly express the decay rate of the related kernel.